Questions tagged [topological-vector-spaces]

A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.

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Is Schauder's conjecture resolved?

Schauder's conjecture: "Every continuous function, from a nonempty compact and convex set in a (Hausdorff) topological vector space into itself, has a fixed point." [Problem 54 in The ...
57319's user avatar
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21 votes
3 answers
3k views

Can you tell whether a space is Banach from the unit ball?

Let $V$ be a real vector space. It is well known that a subset $B\subset V$ is the unit ball for some norm on $V$ if and only if $B$ satisfies the following conditions: $B$ is convex, i.e. if $v,w\...
Jim Belk's user avatar
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20 votes
3 answers
2k views

Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way: Let $X$ be a set and let $\mathcal{P}(...
Bipolar Minds's user avatar
18 votes
7 answers
7k views

Grothendieck on Topological Vector Spaces

In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on Topological Vector Spaces (TVS), apparently, he told Bernard Malgrange ...
16 votes
3 answers
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Why is multiplication on the space of smooth functions with compact support continuous?

I asked the question Why is multiplication on the space of smooth functions with compact support continuous? on M.SE sometime ago but I didn't receive a satisfactory answer. I was reading this ...
Hugo's user avatar
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1 answer
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Classification of non-Hausdorff topological vector spaces

It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean ...
HeinrichD's user avatar
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16 votes
0 answers
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Reference request for Grothendieck's work on "Integration with values in a topological group"

Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
user avatar
15 votes
1 answer
2k views

Bases for spaces of smooth functions

Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all $d\in\...
Neil Strickland's user avatar
14 votes
0 answers
670 views

strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
Chris Wendl's user avatar
13 votes
2 answers
991 views

Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?

Motivation A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
Charlie Cunningham's user avatar
12 votes
1 answer
781 views

Equivalence of σ-convex hull and closed convex hull

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
Gregory D.'s user avatar
12 votes
2 answers
795 views

Nuclear operators/spaces and transfer operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
Felipe Pérez's user avatar
12 votes
1 answer
466 views

Status of the compact AR problem?

The so-called "compact AR Problem" reads: Is every compact convex set in a metrizable topological vector space an absolute retract? It is open according to the chapter by T. Banakh, R. Cauty and ...
Sergey Melikhov's user avatar
11 votes
5 answers
708 views

Colimits in the category of (not necessarily locally convex) topological vector spaces

Do colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general? If no, is there a well-known condition of when they exist? If ...
Junekey Jeon's user avatar
11 votes
2 answers
624 views

Do locally convex topological vector spaces embed into diffeological spaces?

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...
David Roberts's user avatar
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11 votes
1 answer
235 views

Bilinear product of two summable families

Consider the following statement, which I suspect is false as written: Let $E,F,G$ be (Hausdorff) topological vector spaces (over $\mathbb{R}$), let $\varphi\colon E\times F\to G$ be continuous and ...
Gro-Tsen's user avatar
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11 votes
0 answers
200 views

Is it possible to take "limits up to homotopy"?

Before I begin, an apology: it's been a while since I've done much analysis, so I might be misusing (or just missing) terminology. I have a chain complex $(V_\bullet,\partial)$ of topological vector ...
Theo Johnson-Freyd's user avatar
10 votes
2 answers
677 views

non-artificial examples of non-smooth and non-admissible representations of GL_2

Let $F$ be a finite degree extension over $\mathbf{Q}_p$ and consider the locally profinite group $G:=GL_2(\mathbf{Q}_p)$. P1: Give an interesting example (non-artificial one, i.e., one that arises ...
Hugo Chapdelaine's user avatar
10 votes
2 answers
849 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
Abdelmalek Abdesselam's user avatar
10 votes
1 answer
304 views

Closed vector subspaces of large powers of R

By a large power of $\mathbb R$ is meant a topological vector space which is the product of infinitely many copies of the real line. Is every closed subspace of such a TVS linearly homeomorphic to ...
Gene Kaufman's user avatar
10 votes
1 answer
404 views

Meaning of topological tensor products in Frenkel-Gaitsgory

The appendix to http://arxiv.org/abs/math/0508382 by Frenkel & Gaitsgory (following an earlier work of Beilinson) describes three different monoidal structures, denoted by $\otimes^!,\otimes^*,$ ...
Ben G's user avatar
  • 403
10 votes
2 answers
2k views

Pull-back of generalized functions

Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation $f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
asv's user avatar
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9 votes
6 answers
751 views

Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
J. van Dobben de Bruyn's user avatar
9 votes
2 answers
818 views

$C^\infty$-vectors in general representations of Lie groups on locally convex spaces

This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...
Keenan Kidwell's user avatar
9 votes
1 answer
262 views

Is a Banach lattice isomorphic to a Hilbert space in fact a Hilbert lattice?

The title says it all: Let $E$ be a Banach lattice, which is isomorphic to a Hilbert space (as normed spaces). Is there an equivalent Hilbert norm on $E$, which still makes it a Banach lattice with ...
erz's user avatar
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9 votes
0 answers
675 views

Infinite-dimensional affine space in algebraic geometry and algebraic topology

In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
Tim Campion's user avatar
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8 votes
5 answers
2k views

Topological vector space textbook with enough applications

(Sorry for my bad English.) For "applications", I mean applications in math, not real-life. There are many textbooks about topological vector space, for example, GTM269 by Osborne, Modern Methods in ...
8 votes
5 answers
477 views

Reference for : a Fréchet nuclear space is Montel

I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact" Thank you in advance for the help!
Loïc Teyssier's user avatar
8 votes
4 answers
742 views

Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...
Coffee's user avatar
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8 votes
2 answers
568 views

Attempted Banachification of a space

In a recent blog post, Terry Tao mentioned the question of how to tell if a Hausdorff topological vector space admits a finer topological structure which happens to be the topology of a Banach space (...
Will Sawin's user avatar
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8 votes
1 answer
554 views

Two vector spaces with homeomorphic open subsets are isomorphic?

Is it true that if $ E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the ...
usr203050's user avatar
  • 103
8 votes
2 answers
350 views

Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer. If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a ...
J. van Dobben de Bruyn's user avatar
8 votes
1 answer
616 views

When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?

Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form $$ u(A)=\...
Sergei Akbarov's user avatar
8 votes
2 answers
647 views

"Economic" Eilenberg-MacLane topological abelian groups

This might be regarded as a sequel to my previous "Economic" CW-structure for Eilenberg-MacLane spaces? However the content seems to be quite different. I believe it is easy to prove that ...
მამუკა ჯიბლაძე's user avatar
8 votes
1 answer
489 views

Examples of topologies compatible with a given dual pair

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...
Tobias Diez's user avatar
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8 votes
1 answer
510 views

Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?

Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial. For $m\in \mathbb{Z}$, we also define $\...
Goulifet's user avatar
  • 2,174
8 votes
1 answer
666 views

Integration with values in a topological vector space

Is there a general theory of integration of functions with values in a topological vector space (not necessarily locally convex)? Browsing through mathoverflow posts, I came across a discussion ...
TVS_integration's user avatar
8 votes
0 answers
183 views

On "linearly independent" metric spaces

Urysohn's universal metric space $\Bbb U$ satisfies the following surprising property: Whenever $i\colon\Bbb U\to E$ is an isometric embedding into a normed vector space such that $0\not\in i(\Bbb U)$...
Alessandro Codenotti's user avatar
8 votes
0 answers
437 views

When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra

For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ? More precisely, do we have ...
LCO's user avatar
  • 496
7 votes
1 answer
466 views

closed subspaces of locally convex inductive limits

It's a duplicate of this question, since I really want to get an explanation. Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex ...
user35953's user avatar
  • 183
7 votes
1 answer
667 views

Topological groups in which all subgroups are closed

General question: does there exist a nondiscrete topological group $G$ such that all subgroups of $G$ are closed? Or, does there exist a nondiscrete topological vector space $V$ such that all vector ...
Leonid Positselski's user avatar
7 votes
1 answer
371 views

Is any dual metrizable locally convex space a Frechet space?

[I have posted this question on MSE some time ago, but received no answer.] The title basically says all of it. If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I ...
erz's user avatar
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7 votes
3 answers
2k views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
SBF's user avatar
  • 1,605
7 votes
1 answer
189 views

$GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?

Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
Giulia's user avatar
  • 73
7 votes
1 answer
1k views

Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
user68620's user avatar
7 votes
0 answers
205 views

Simultaneous Strong Law of Large Number classes?

Say that $C$ is a SSLLN class of subsets of some topological space $V$ provided that for every sequence of i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
Alexander R Pruss's user avatar
6 votes
3 answers
1k views

Topological vector spaces that are isomorphic to their duals

After reviewing the (locally convex) topological vector spaces that I know, the only examples I could find where there is an isomorphism from the space to its (anti)dual, are Hilbert spaces. So my ...
Oliver's user avatar
  • 347
6 votes
3 answers
2k views

Sequential topological vector spaces

Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
Johannes Hahn's user avatar
6 votes
3 answers
486 views

Proof of the Schauder Lemma

Schauder's Lemma in functional analysis states the following: Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\...
Dominic Wynter's user avatar
6 votes
4 answers
1k views

On locally convex (and compactly generated) topological vector spaces

Part 1: How big is the category $TVS_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)? In other words (and less cheekily), is there a free locally convex TVS having any ...
David Roberts's user avatar
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